$\begingroup$Is your question how to prove that $M_t-B_t$ has the same distribution as $\left| B_t\right|$ for each $t\geq 0$?$\endgroup$
– peerApr 10 '17 at 11:22

$\begingroup$@peer The formulation of the question clearly points at the identity in distribution of the processes, not only of the random variables as in your answer.$\endgroup$
– DidApr 10 '17 at 13:26